Fyodorov, Y V and Simm, N J (2016) On the distribution of maximum value of the characteristic polynomial of GUE random matrices. Nonlinearity, 29. pp. 2837-2855. ISSN 0951-7715
![]() |
PDF
- Published Version
Available under License Creative Commons Attribution. Download (1MB) |
Abstract
Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random N×N matrices H from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of DN(x):=−log|det(xI−H)| as N→∞ and x∈(−1,1). We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of DN(x).
Item Type: | Article |
---|---|
Schools and Departments: | School of Mathematical and Physical Sciences > Mathematics |
Subjects: | Q Science > QA Mathematics |
Depositing User: | Nicholas Simm |
Date Deposited: | 13 Jun 2018 10:03 |
Last Modified: | 13 Jun 2018 10:03 |
URI: | http://srodev.sussex.ac.uk/id/eprint/76461 |
View download statistics for this item
📧 Request an update